Mathematics, 11.12.2020 04:30 BeautyxQueen

A conical container, with vertex down, has a height of 6 cm and a diameter of 2 cm. It is leaking water at the rate of 1 cubic centimeter per minute. Find the rate at which the water level h is dropping when h = 3 cm.

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Answer from: carley608

$\displaystyle \frac{dh}{dt}=-\frac{4}{\pi}\approx-1.2732\text{ centimeters per minute}$

The water level is dropping by approximately 1.27 centimeters per minute.

Step-by-step explanation:

Please refer to the attached diagram.

The height of the conical container is 6 cm, and its radius is 1 cm.

The container is leaking water at a rate of 1 cubic centimeter per minute.

And we want to find the rate at which the water level h is dropping when the water height is 3 cm.

Since we are relating the water leaked to the height of the water level, we will consider the volume formula for a cone, given by:

$\displaystyle V=\frac{1}{3}\pi r^2h$

Now, we can establish the relationship between the radius r and the height h. At any given point, we will have two similar triangles as shown below. Therefore, we can write:

$\displaystyle \frac{1}{6}=\frac{r}{h}$

Solving for r yields:

$\displaystyle r=\frac{1}{6}h$

So, we will substitute this into our volume formula. This yields:

$\displaystyle \begin{aligned} V&=\frac{1}{3}\pi \Big(\frac{1}{6}h\Big)^2h\\ &=\frac{1}{108}\pi h^3\end{aligned}$

Now, we will differentiate both sides with respect to time t. Hence:

$\displaystyle \frac{d}{dt}[V]=\frac{d}{dt}\Big[\frac{1}{108}\pi h^3\Big]$

The left is simply dV/dt. We can move the coefficient from the right:

$\displaystyle \frac{dV}{dt}=\frac{1}{108}\pi\frac{d}{dt}\big[h^3\big]$

Implicitly differentiate:

$\displaystyle\begin{aligned} \frac{dV}{dt}&=\frac{1}{108}\pi(3h^2\frac{dh}{dt})\\ &=\frac{1}{36}\pi h^2\frac{dh}{dt}\end{aligned}$

Since the water is leaking at a rate of 1 cubic centimeter per minute, dV/dt=-1.

We want to find the rate at which the water level h is dropping when the height of the water is 3 cm.. So, we want to find dh/dt when h=3.

So, by substitution, we acquire:

$\displaystyle -1=\frac{1}{36}\pi(3)^2\frac{dh}{dt}$

Therefore:

$\displaystyle -1=\frac{1}{4}\pi\frac{dh}{dt}$

Hence:

$\displaystyle \frac{dh}{dt}=-\frac{4}{\pi}\approx-1.2732\text{ centimeters per minute}$

The water level is dropping at a rate of approximately 1.27 centimeters per minute.

A conical container, with vertex down, has a height of 6 cm and a diameter of 2 cm. It is leaking wa

Answer from: Quest

-3

step-by-step explanation:

a drawing really does here:

so i draw the horizontal line y=2 and the vertical line x=-2 at this intersection i put b (this is where my right angle is going to be). so point b is at (-2,2). point a is 5 units long and is also on y=2 so this point has to be (3,2) or (-7,2) -doesn't matter i will go with (3,2).

now we know c is at (-2,y) we don't know the y yet bet we know that c is on x=-2.

we have the area of the triangle is .5bh=12.5

we know the base length is 5 (ab)

we have the height is (2-y) if you look at your

plug in

.5(5)(2-y)=12.5

solve for y to get the y-coordinate for c

2.5(2-y)=12.5

divide both sides by 2.5 giving:

2-y=5

now subtract 2 on both sides

-y=3

y=-3

Answer from: Quest

answer is 12345678900987654321234567890987654321

step-by-step explanation:

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A conical container, with vertex down, has a height of 6 cm and a diameter of 2 cm. It is leaking wa...